# Bounded loss function whose minimizer is the mean

Let

$$y = \operatorname*{argmin}_\hat{x} \operatorname*{E}_x L(|\hat{x} - x|)$$

where $$L$$ is a loss function. As noted here, if $$f(s) = s^p$$ then

• $$p \rightarrow 0$$ implies $$y \rightarrow$$ the mode
• $$p = 1$$ implies $$y =$$ the median
• $$p = 2$$ implies $$y =$$ the mean
• $$p \rightarrow \infty$$ implies $$y \rightarrow$$ the midrange

Is there an $$L$$ bounded on $$[0,\infty)$$ whose minimizer $$y$$ is also the mean?

• My gut: not without some constraints on the range of $x$ or its residuals. And cutting the loss function off at the biggest possible $x-\hat{x}$ seems like a cheater's answer. – AdamO Oct 23 '19 at 16:14